Tiamat is one of the four Fiends (Chaoses) and the boss of the Sky Castle. Then you'll fight a stronger Tiamat on the way through the final dungeon.

STRATEGY

The first time you fight Tiamat there are 2 ways to win. You could use FAST, fight strong, and hit hard and beat away at her 1000HP. The other way is her weakness to the Poison element. Cast BANE or BRAK enough times and eventually it should work. The Bane Sword is helpful in this process. Sometimes it'll take longer than the traditional way though. The second time you face Tiamat she doesn't have a weakness, so you'll have to fight.

MYTH/ORIGIN/OTHER GAMES

Tiamat has the same name in every version. Origin of Tiamat courtesy of Kyle Cox: "She is a humungous female dragon that inhabits the ocean of Chaos in Babalonyan lore. She is the mother of all creation, including the gods, and is also the personification of saltwater. Her consort is Apsu, the yang to her yin. He is the massive freshwater abyss beneath the Earth. This union is a strange one indeed, but according to Babalonyan lore, the union of salt and fresh waters caused the birth of the first two gods, Lachmu and Lachamu. Tiamat was instrumental in the creation of what we know as the Earth, though she herself did not will it. One of the later-generations of gods, Marduk (born of fresh water), killed her by cleaving her in half. He used her upper half to create the sky, and her lower half became the Earth. Her tears became the source of the Tigris and Euphrates rivers, and her water became the clouds." She's also found in Dungeons & Dragons, and was in at least one episode of the 80's D&D cartoon show. Tiamat returns as a boss in FF2, as Ultimecia's summon in FF8, and as a Chaos in FF9.